Integrand size = 21, antiderivative size = 136 \[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=-\frac {96 \sqrt {a+i a \sinh (e+f x)}}{f^4}-\frac {12 x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {48 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \]
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Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3400, 3377, 2718} \[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=-\frac {96 \sqrt {a+i a \sinh (e+f x)}}{f^4}+\frac {48 x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f^3}-\frac {12 x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f} \]
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Rule 2718
Rule 3377
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (\text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int x^3 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx \\ & = \frac {2 x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}-\frac {\left (6 \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int x^2 \cosh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f} \\ & = -\frac {12 x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}-\frac {\left (24 i \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f^2} \\ & = -\frac {12 x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {48 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}-\frac {\left (48 \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \cosh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f^3} \\ & = -\frac {96 \sqrt {a+i a \sinh (e+f x)}}{f^4}-\frac {12 x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {48 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \\ \end{align*}
Time = 2.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {2 \left (i \left (48 i+24 f x+6 i f^2 x^2+f^3 x^3\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (-48 i+24 f x-6 i f^2 x^2+f^3 x^3\right ) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+i a \sinh (e+f x)}}{f^4 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 0.55 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) {\mathrm e}^{-f x -e}}\, \left (i x^{3} f^{3}+{\mathrm e}^{f x +e} f^{3} x^{3}+6 i x^{2} f^{2}-6 f^{2} x^{2} {\mathrm e}^{f x +e}+24 i x f +24 f x \,{\mathrm e}^{f x +e}+48 i-48 \,{\mathrm e}^{f x +e}\right ) \left ({\mathrm e}^{f x +e}-i\right )}{\left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) f^{4}}\) | \(151\) |
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Exception generated. \[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=\int x^{3} \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}\, dx \]
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\[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{3} \,d x } \]
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\[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{3} \,d x } \]
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Time = 1.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93 \[ \int x^3 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{e+f\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-e-f\,x}\,{\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}\,\left (f^3\,x^3\,{\mathrm {e}}^{e+f\,x}+f\,x\,24{}\mathrm {i}+f^2\,x^2\,6{}\mathrm {i}+f^3\,x^3\,1{}\mathrm {i}-6\,f^2\,x^2\,{\mathrm {e}}^{e+f\,x}-48\,{\mathrm {e}}^{e+f\,x}+24\,f\,x\,{\mathrm {e}}^{e+f\,x}+48{}\mathrm {i}\right )}{f^4\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \]
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